Department of Physics

Seminar Ib

Thermal radiation heat transfer between surfaces

Author: Luka Klobučar

Adviser: prof. dr. Iztok Tiselj

Co-adviser: dr. Boštjan Končar

Ljubljana, February 2016

Abstract

Numerical simulations of thermal radiation heat transfer are crucial to prepare the DEMO fusion

reactor conceptual design. Despite the relatively simple problem numerical simulations contains

significant errors. In order to reduce these error it important to understand the thermal radiation heat

transfer theory. In this seminar we will roughly describe models for prediction of thermal radiation

heat transfer between surfaces based on two assumptions: surfaces form ideal closed enclosure and are

separated by nonparcipating media. The portion of radiation exchanged between two differently

oriented surfaces has to be defined by a geometric function known as view factor, which is developed

for gray diffusely radiating surfaces. At the end of the seminar a simple 3D furnace cavity example is

presented.

2

Contents

2.1 Conduction ............................................................................................................................................ 3

2.2 Convection ............................................................................................................................................. 3

2.3 Thermal Radiation heat transfer ............................................................................................................ 4

4.1 View factor relations ............................................................................................................................. 7

4.2 Methods for the evaluation of view factor ............................................................................................. 7

1 Introduction

Thermal radiation heat transfer is one of fundamental modes of heat transfer. All matter at a nonzero

absolute temperature emit electromagnetic waves which are called thermal radiation. Thus, the heat

transfer by thermal radiation is everywhere around us. Human body is much more sensitive to

conduction and convection, however we can feel the radiation too. Conduction and convection usually

play an important role in heat transfer under normal conditions (small temperature difference and

participating media). With the development of new technologies (space applications, thermal and

nuclear power plant, vacuum applications, solar energy, etc) that operate at extreme conditions (large

temperature difference and vacuum) thermal radiation becomes important and received increasing

attention and development.

The motivation for this seminar is thermal analysis of the future DEMO fusion reactor that will

operate at extreme temperature condition and at higher vacuum. The DEMO tokamak is composed of

many systems operating at very different temperatures ideally separated by vacuum. In fact, typical

pressure inside the vacuum vessel is in the range between 10-7

to 10-8

mbar. On Fig. 1 major

components and their operational temperatures are presented. In a very simplified analysis of DEMO

tokamak the direct heating of in-vessel components by escaped plasma particles and deposited internal

heat by neutrons are not modeled. Instead their contribution is converted to different, but fixed

temperatures of the components taking into account that they are actively cooled. In high vacuum the

heat transfer by convection doesn’t play a significant role, therefore thermal radiation remains the

prevailing heat transfer mechanism at such conditions.

1 Introduction .................................................................................................................................................. 2

2 Heat Transfer mechanisms .......................................................................................................................... 3

3 Radiation exchange between surfaces ......................................................................................................... 4

4 View factor ............................................................................................................................................ 5

5 Calculation example ..................................................................................................................................... 8

6 Conclusion ................................................................................................................................................... 12

References ............................................................................................................................................................ 12

3

Heat loads in the DEMO reactor system can be calculated by the means of numerical simulation.

However, the main objective of this seminar is understanding the main heat radiation mechanism

through the theoretical analysis. Theoretical background of calculating heat transfer is given and

analytical solution for a simple example is presented. In addition, the parameters and relations used to

check the accuracy of numerical solution are introduced. In the simulation of thermal radiation the

calculation of the view factors is the most important for the accuracy of the calculations. So it is

important to understand the view factor method theory.

cryostat (CRY) (300 K)

magnets (4 K)

vacuum vessel (VV) (473 K)

blankets (573 K)

VV thermal shield (80 K)

divertor (573 K)

CRY thermal shield (80 K)

Figure 1: 1/18th (20°) section of the entire DEMO tokamak model [5]

2 Heat Transfer mechanisms

Heat transfer is the exchange of thermal energy between systems with different temperatures. There

are different modes of heat transfer: conduction, convection and thermal radiation depending on the

state of systems.

2.1 Conduction

Conduction is a mode of the heat transfer when temperature gradient exists in a stationary solid or

fluid medium. Energy is transferred from warmer to colder parts of the media by interactions between

them. Heat conduction is expressed by Fourier’s law:

(1)

where is area and is the temperature gradient. The parameter

is thermal

conductivity and is characteristic property of the material. [1]

2.2 Convection

Convection is a mode of heat transfer that occurs between a surface and a moving fluid when they are

at different temperatures. Heat is transferred from one place to another by the movement of fluids.

Convection heat transfer may be classified according to the nature of the flow. Free (natural)

convection occurs when the fluid motion is caused by buoyancy forces and forced convection occurs

when the fluid flow is caused by external source, such as fan, pump, etc. Whatever the nature of the

flow, the process described by Newton’s law of heat transfer.

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(2)

where is area and is the difference between the surface and fluid temperatures. The

parameter

is convective heat transfer coefficient – an empirical correlation that depends on

the surface geometry, physical properties of the fluid and flow properties. Values of have been

measured and tabulated for typical fluids and flow regimes. [1]

2.3 Thermal Radiation heat transfer

Thermal radiation is a mode of the heat transfer between two surfaces at different temperatures in the

absence of media. Electromagnetic waves do not need matter to propagate. Even better, they are most

efficiently propagated in vacuum. Total emissive power of the blackbody is prescribed by the Stefan-

Boltzmann’s law

(3)

where is absolute temperature,

is the Stefan-Boltzmann constant and

is flux. The surface may be subjected to radiation from its surroundings which is in general

treated as a gray surface.

(4)

where is the temperature of the surface, is the temperature of surroundings and is

emissivity, coefficient defining how efficiently surface emits energy relative to the blackbody. [1]

By comparing all modes of heat transfer two essential differences can be noticed. Both conduction and

convection require the presence of medium to transfer the energy. On the other hand, radiation doesn’t

require a medium for transfer the energy. Another difference arises from temperature dependence.

Conduction and convection heat transfer are roughly linearly proportional to temperature difference in

the case there is no strong dependence of and . Radiative heat transfer is proportional to

temperature to the fourth power. Thus, radiative heat transfer becomes more important at higher

temperature differences. Vacuum and high temperature differences are characteristics that describe the

DEMO fusion reactor conditions very well.

3 Radiation exchange between surfaces

The theory for radiation exchange between surfaces described in the present seminar is based on two

assumptions. Surfaces form an enclosure and surfaces are separated by a nonparticipating media.

Radiatively nonparticipating media has no effect on the transfer of radiation between surfaces. There

is no scattering, no emission and no absorbtion. Such medium is vacuum, and also other monatomic

and most diatomic gases at low and moderate temperatures, at temperatures before ionization and

dissipation occurs. In fact, in many engineering applications media doesn’t affect the radiation heat

transfer.

The energy balance on the opaque surface is: . Absorption depends on

irradiation, which depends on emission from other surfaces including those far away from the

observed surface. To make total radiative energy balance we must consider the entire enclosure, which

is assumed to be closed. Thus, all radiation contributions are accounted for. An open enclosure is in

practice closed by introducing artificial surfaces. For example, opening can be considered as a surface

with zero reflectivity and as a radiation source when presenting environmental radiation. Enclosure

5

may be composed of complex geometries which can bring lots of difficulties in calculations.

Therefore, the enclosure may be idealized by inventing alternative simple surfaces and by assuming

surfaces to be isothermal with constant (average) heat flux values across them, as indicated in Fig. 2.

[1], [2], [3]

Figure 2: Real and idealized enclosure.

Radiation exchange between surfaces in addition to their radiative properties and temperatures

strongly depends on the surfaces geometries, orientations and separations distance. This all leads to

development of geometric function known as view factor.

4 View factor

View factor can be also called configuration factor, shape factor, form factor. View factor is

dimensionless factor that determines how much of a surface is visible to another surface and is a pure

geometric property. Definition: The view factor is defined as the fraction of the radiation leaving

surface that is intercepted by surface .

(5)

(a) (b) I

Figure 3: (a)Radiation exchange between two surfaces, (b)interpretation of solid angle, (c) geometry

used to obtained emissive power

The rate at which radiation leaves and is intercepted by may be expressed as:

(6)

Ti

Tn

T1

T2

T r

6

Where is the intensity of radiation leaving surface by emission and reflection and is

solid angle subtended by when viewed from . Solid angle is defined as the projection of the

surface onto a plane normal to the direction vector, divided by the distance squared (see Fig. 3b)

(7)

Radiative intensity is defined as radiative energy flow per unit solid angle and unit area normal

to the rays.

(8)

From Eq. 6 it follows that:

(9)

Total emitted energy from into hemisphere above per unit surface area is (see Fig. 3c):

(10)

Similarly can be written for radiosity, which is total radiative energy leaving from into hemisphere

above per unit surface area. In general radiosity includes both emissions and reflections.

(11)

Assuming that surface emits and reflects diffusely it follows:

(12)

The total rate at which radiations leaves surface and is intercepted by may then be obtained

by integration over the two surfaces. Assuming that the radiosity is uniform over the surface

(13)

From the definition of the view factor (Eq. 5) it follows that:

(14)

Similarly, the view factor is defined

(15)

This equation may be used to determine the view factor associated with any two surfaces that are

diffuse emitters and diffuse reflectors and have uniform radiosity. [2]

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4.1 View factor relations

The view factor also satisfies two useful relations.

Reciprocity relations (law of reciprocity)

From (Eg. 14) and (Eg. 15) it follows:

(16)

Summation relation (summation rule)

Summation relation follows directly from the definition of the view factor. From the requirement that

the enclosure must be closed it follows that sum of all view factors must be summed up in one.

Summation relation for enclosure of surfaces is:

(17)

In the sum is also view factor , which needs special consideration. If the plane is convex, no

radiation leaving will strike itself so . If the surface is concave, it sees itself and part of

radiation leaving will be intercepted by itself so .

These two relations aren’t just for determining the analytical solution but are also important for the

verification of the numerical solution. Reciprocity relation is used to check the accuracy of individual

view factor and law of conservation of energy is verified with the summation rules. [1], [2]

To calculate radiation exchange in the enclosure of surfaces a total of view factors are required.

The view factors on surfaces can be written in matrix form:

(18)

However, all view factors do not need to be calculated directly. Most of them can be calculated by

using view factor relations. view factors may be obtained from the equations of the summation

relations and

view factors may be obtained from the reciprocity relations. When the enclosure

in the model is divided into surfaces that can’t sees itself so , the total number of view

factors, that must be calculated directly is then:

(19)

So

view factor must be calculated directly. In some cases, the use of symmetry can

additionally reduce the number of view factor that must be calculated directly. [2], [3]

4.2 Methods for the evaluation of view factor

The calculation of radiative view factor between any of two finite surfaces requires solving of the

double area integral, or fourth-order integration. Such integrals are difficult to evaluate analytically

except for very simple geometries. For more complicated geometries, the view factors are calculated

by numerical integration. Such integrations can be computationally expensive, depending on the

complexity of the geometry. Because of that, mathematical methods were developed for evaluation of

the view factor. There are several tables and charts for frequently used geometries. The most complete

tabulation is given in a catalogue by Howell [4]. A shorter list of integration methods is provided in

[1]. In this seminar direct surface integration and inside sphere method are briefly presented.

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5 Calculation example

Knowing the theory of view factors, a simple example can be calculated.

A furnace cavity, which is in the form of a cylinder with diameter and height of (see

Fig. 5b), is open at one end to large surroundings that are at . The bottom of the cavity is

heated independently, as the side of the cavity. All interior surfaces of the cavity may be approximated

as blackbodies and are maintained at . The backs of the electrically heated surfaces are

well insulated. What is the radiation heat loss from the furnace? [2]

Assumptions:

- Interior surfaces behave as blackbodies with uniform radiosity and irradiation.

- Heat transfer by convection is negligible.

- Backs of electrically heated surfaces are adiabatic (no heat transfer).

Furnace is an open cylinder and includes bottom surface ( ) and side surface ( ,

). Since the enclosure must be closed, an artificial surface which represents the opening

has to be introduced ( ). The irradiation from the surroundings is equal to emission

from a blackbody at . Thus, the top surface is considered as a black surface at

. The radiation heat loss from the furnace is radiation from the furnace bottom and side walls to

the top surface.

So three surfaces are considered. To calculate radiation exchange nine view factors are needed. Let’s

see how many view factors need to be calculated directly. Now we know that

view factor

must be calculated directly. We have two flat ( ) and one concave surface

( ) so . In our example we must directly calculated just one view factor, others

are obtained from the view factor relations.

(20)

Of course, we may also consider the symmetry.

, , and (21)

The easiest way to calculate the view factor is to look into the catalogues and hope that you could find

it for your geometries. In the catalogue [4], we can find the view factors for the geometry in Fig 5.

(a) (b)

Figure 5:Configuration for (a) disk to parallel coaxial disk of unequal radius, (b) circular cylinder

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Disk to parallel coaxial disk of unequal radius (C-41) [4] (see Fig. 5a)

Definitions:

Equation:

In our case so it follows:

If

it follows:

(22)

Now when we have the view factor we can calculate the others. But let’s see which view factor can

be found in the tables [4] for our example.

Base of right circular cylinder to inside surface of cylinder (C-79) [4] (see Fig. 5b)

Equation:

(23)

Inside surface of right circular cylinder to itself (C-78) [4] (see Fig. 5b)

Equation:

(24)

In our case, all factors can be obtained from the catalogs, but this is not always the case. As an

example let’s try to get analytical solutions for the view factor and compare it with solution in

tables (Eg. 25).

Surface integration [1]

To evaluate the equation (Eg. 14), the integrand must be known in terms of local coordinate system

that describes the geometry of the two surfaces as shown in Fig. 6. Using an arbitrary coordinate

origin, a vector pointing from origin to a point on a surface may be written as:

(25)

where , and are unit vectors. The vector from to may be written as:

(26)

And the length of this vector is:

(27)

Local surface normals are

(28)

so and may be expressed as:

(29)

Now all integrands have been expressed and our example can be solved.

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Figure 6:Configuration for the view factor between parallel, coaxial disks of unequal radius

View factor between the two parallel, coaxial disk of radius and can be written as:

First transform Cartesian coordinates to cylindrical coordinates (see Fig. 6) and express integrand.

(30)

When all is inserted in equation for view factor is obtained:

(31)

This integral can be solved numerically, but the analytical solution is very difficult. Let’s try another

method. Geometry in this case is suitable for special method called inside-sphere method.

The inside-sphere method [1]

Consider two surfaces and that are both parts of the surface of same sphere as shown in Fig. 7a.

(a) (b)

Figure 7: Configuration for (a) the inside-sphere method, (b) view factor between coaxial parallel

disks.

Ai

A j

R

R

S

dA i

dA j

i

j

A1 R1

A3 R3

h

R

R

A1'

A3'

1

3

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We can see that and . From relations (see Eg. 15) it follows that:

(32)

Therefore, because of the unique geometry of a sphere the view factor between two surfaces on the

same sphere only depends on the size of the receiving surface ( ) and not on the location of either

one. The inside-sphere method could be used to find out the view factor between two surfaces that

may not necessarily lie on a sphere. This method can be used to calculate some view factors for our

example geometry.

View factor between two parallel, coaxial disks of unequal radius.

As shown in Fig. 7b we see that it is possible to place the disks inside the sphere in such way that

entire peripheries of both disk lie on the surface of the sphere. Since all radiation from to travels

on to the spherical cap , and since all radiation from to must pass through , we have

. And similar for .

(33)

The areas of the spherical caps are calculated as:

(34)

So that

(35)

From Fig. 7b it can be seen that

, which results in:

(36)

In our case and

so it follows:

(37)

And it is the same as that found in the tables (Eg. 25). Similar analytical integrations would obtain

view factor (Eg. 26). The rest of the view factors are obtained by using the view factors relations, and

this gives a table of all view factors:

Table I. All analytical expressed view factors

Now we could calculate the heat loss from the furnace through the opening. Net radiative exchange

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between two surfaces is:

(38)

Net radiative transfer from surface is:

(39)

So the radiation heat loss through the open surface is:

(40)

6 Conclusion

This seminar briefly summarizes the main heat transfer mechanisms and focus on thermal radiation

heat transfer exchange between surfaces. Thermal radiation propagates in vacuum thus radiative heat

transfer is a long-range phenomenon. The energy balance of the observed surface is therefore affected

by emission from the distant surface. To calculate total radiative energy balance the entire enclosure

must be consider. Radiation exchange between surfaces in addition to their radiative properties and

temperatures strongly depends on the surfaces geometries, orientations and separations distance. This

all led to development of geometric function known as view factor. In this seminar the view factor for

gray diffusely radiating surfaces is presented. View factor relations and methods for their calculations

are described. Finally the methods for view factor calculations were demonstrated on the selected

example.

Most engineering cases may often be approximated with such ideal gray and diffusely radiating

surfaces in closed enclosure so the view factor method is often used. Numerical calculation of the view

factor is the key point in thermal radiation calculations. As we know the calculations of the view factor

could be very complex and could bring numerical errors in the heat transfer calculation. Therefore, it

is important to understand the view factor method theory.

References

[1] M. F. Modest, Radiative heat transfer, (McGraw-Hill, New York, 1993)

[2] T. L. Bergman, A. S. Lavine, F. P. Incropera, and D. P. Dewitt, Fundamentals of heat and mass

transfer, (John Wiley & Sons, Hoboken, 2011)

[3] R. Siegel and J. R. Howell, Thermal radiation heat transfer, (Taylor & Francis, Washington,

1992)

[4] J. R. Howell, “A catalog of radiation heat transfer configuration factors”,

http://www.thermalradiation.net/indexCat.html, (4. 2. 2016)

[5] B. Končar, M. Draksler, O. Costa Garrido, I. Vavtar, Thermal analysis of DEMO tokamak

2015, (Jožef Stefan Institute, 2016)